December  2020, 25(12): 4801-4822. doi: 10.3934/dcdsb.2020127

Small time asymptotics for SPDEs with locally monotone coefficients

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

* Corresponding author: Wei Liu

Received  August 2019 Published  April 2020

Fund Project: The research of W. Liu is supported by NSFC (No. 11822106, 11831014, 11571147), the research of Y. Xie is supported by NSFC (No. 11771187, 11931004) and PAPD of Jiangsu Higher Education Institutions

This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous medium equations, stochastic $ p $-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.

Citation: Shihu Li, Wei Liu, Yingchao Xie. Small time asymptotics for SPDEs with locally monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4801-4822. doi: 10.3934/dcdsb.2020127
References:
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show all references

References:
[1]

H. Abdallah, A Varadhan type estimate on manifolds with time-dependent metrics and constant volume, J. Math. Pures Appl., 99 (2013), 409-418.  doi: 10.1016/j.matpur.2012.09.007.  Google Scholar

[2]

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[3]

S. Aida and T. S. Zhang, On the small time asymptotics of diffusion processes on path groups, Potential Anal., 16 (2002), 67-78.  doi: 10.1023/A:1024868720071.  Google Scholar

[4]

T. Ariyoshi and M. Hino, Small-time asymptotic estimates in local Dirichlet spaces, Electron. J. Probab., 10 (2005), 1236-1259.  doi: 10.1214/EJP.v10-286.  Google Scholar

[5]

M. AvellanedaD. Boyer-OlsonJ. Busca and P. Friz, Application of large deviation methods to the pricing of index options in finance, C. R. Math. Acad. Sci. Paris, 336 (2003), 263-266.  doi: 10.1016/S1631-073X(03)00032-3.  Google Scholar

[6]

H. BerestyckiJ. Busca and I. Florent, Computing the implied volatility in stochastic volatility models, Comm. Pure Appl. Math., 57 (2004), 1352-1373.  doi: 10.1002/cpa.20039.  Google Scholar

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Z. Brzézniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295.  doi: 10.1080/17442509708834122.  Google Scholar

[8]

Z. Brzézniak and S. Peszat, Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process, Studia Math., 137 (1999), 261-299.  doi: 10.4064/sm-137-3-261-299.  Google Scholar

[9]

Z. BrzeźniakW. Liu and J. H. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310.  doi: 10.1016/j.nonrwa.2013.12.005.  Google Scholar

[10]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61.   Google Scholar

[11]

A. BudhirajaP. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420.  doi: 10.1214/07-AOP362.  Google Scholar

[12]

Y. ChenH. J. Gao and L. L. Fan, Well-posedness and the small time large deviations of the stochastic integrable equation governing short-waves in a long-wave model, Nonlinear Anal. Real World Appl., 29 (2016), 38-57.  doi: 10.1016/j.nonrwa.2015.10.009.  Google Scholar

[13]

Z.-Q. ChenS. Z. Fang and T. S. Zhang, Small time asymptotics for Brownian motion with singular drift, Proc. Amer. Math. Soc., 147 (2019), 3567-3578.  doi: 10.1090/proc/14511.  Google Scholar

[14]

P. L. Chow, Large deviation problem for some parabolic Itô equations, Comm. Pure Appl. Math., 45 (1992), 97-120.  doi: 10.1002/cpa.3160450105.  Google Scholar

[15]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.  Google Scholar

[16]

E. A. Coayla-Teran, P. M. Dias de Magalhães and J. Ferreira, Existence of optimal controls for SPDE with locally monotone coefficients, International J. Control, (2018). doi: 10.1080/00207179.2018.1508849.  Google Scholar

[17]

B. Davis, On the $L^p$-norms of stochastic integrals and other martingales, Duke Math. J., 43 (1976), 697-704.  doi: 10.1215/S0012-7094-76-04354-4.  Google Scholar

[18]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, MA, 1993.  Google Scholar

[19]

Z. Dong and R. Zhang, On the small-time asymptotics of 3D stochastic primitive equations, Math. Methods Appl. Sci., 41 (2018), 6336-6357.  doi: 10.1002/mma.5142.  Google Scholar

[20]

S. Fang and T. S. Zhang, On the small time behavior of Ornstein-Uhlenbeck processes with unbounded linear drifts, Probab. Theory Related Fields, 114 (1999), 487-504.  doi: 10.1007/s004400050232.  Google Scholar

[21]

J. FengJ.-P. Fouque and R. Kumar, Small-time asymptotics for fast mean-reverting stochastic volatility models, Ann. Appl. Probab., 22 (2012), 1541-1575.  doi: 10.1214/11-AAP801.  Google Scholar

[22]

M. Forde and A. Jacquier, Small-time asymptotics for an uncorrelated local-stochastic volatility model, Appl. Math. Finance, 18 (2011), 517-535.  doi: 10.1080/1350486X.2011.591159.  Google Scholar

[23]

M. FordeA. Jacquier and R. Lee, The small-time smile and term structure of implied volatility under the Heston model, SIAM J. Financial Math., 3 (2012), 690-708.  doi: 10.1137/110830241.  Google Scholar

[24]

J. Frehse and M. Růžička, Non-homogeneous generalized Newtonian fluids, Math. Z., 260 (2008), 355-375.  doi: 10.1007/s00209-007-0278-1.  Google Scholar

[25]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften, 260. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9.  Google Scholar

[26]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.  Google Scholar

[27]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dynam. Differential Equations, 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.  Google Scholar

[28]

B. GessW. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.  Google Scholar

[29]

B. Gess, W. Liu and A. Schenke, Random attractors for locally monotone stochastic partial differential equations, J. Differential Equations, 268 (2020), In press. doi: 10.1016/j.jde.2020.03.002.  Google Scholar

[30]

B. L. GuoC. X. Guo and J. J. Zhang, Martingale and stationary solutions for stochastic non-Newtonian fluids, Differential Integral Equations, 23 (2010), 303-326.   Google Scholar

[31]

M. Hino and K. Matsuura, An integrated version of Varadhan's asymptotics for lower-order perturbations of strong local Dirichlet forms, Potential Anal., 48 (2018), 257-300.  doi: 10.1007/s11118-017-9634-x.  Google Scholar

[32]

M. Hino and J. A. Ramirez, Small-time Gaussian behaviour of symmetric diffusion semigroup, Ann. Probab., 31 (2003), 1254-1295.  doi: 10.1214/aop/1055425779.  Google Scholar

[33]

T. Jegaraj, Small time asymptotics for stochastic evolution equations, J. Theoret. Probab., 24 (2011), 756-788.  doi: 10.1007/s10959-010-0336-1.  Google Scholar

[34]

N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, Stochastic Differential Equations: Theory and Applications, 1–69, Interdiscip. Math. Sci., Vol. 2, World Sci. Publ., Hackensack, NJ, 2007. doi: 10.1142/9789812770639_0001.  Google Scholar

[35]

O. A. Ladyzhenskaya, New equations for the description of the viscous incompressible fluids and solvability in large of the boundary value problems for them, Volume V of Boundary Value Problems of Mathematical Physics, Amer. Math. Soc., Providence, 1970. Google Scholar

[36]

S. H. LiW. Liu and Y. C. Xie, Large deviations for stochastic 3D Leray-$\alpha$ model with fractional dissipation, Commun. Pure Appl. Anal., 18 (2019), 2491-2510.  doi: 10.3934/cpaa.2019113.  Google Scholar

[37]

H. Liu and C. F. Sun, On the small time asymptotics of stochastic non-Newtonian fluids, Math. Methods Appl. Sci., 40 (2017), 1139-1152.  doi: 10.1002/mma.4041.  Google Scholar

[38]

W. Liu, Harnack inequality and applications for stochastic evolution equations with monotone drifts, J. Evol. Equ., 9 (2009), 747-770.  doi: 10.1007/s00028-009-0032-8.  Google Scholar

[39]

W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim., 61 (2010), 27-56.  doi: 10.1007/s00245-009-9072-2.  Google Scholar

[40]

W. Liu and M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259 (2010), 2902-2922.  doi: 10.1016/j.jfa.2010.05.012.  Google Scholar

[41]

W. Liu and M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions, J. Differential Equations, 254 (2013), 725-755.  doi: 10.1016/j.jde.2012.09.014.  Google Scholar

[42]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[43]

W. LiuM. Röckner and J. L. da Silva, Quasi-linear (stochastic) partial differential equations with time-fractional derivatives, SIAM J. Math. Anal., 50 (2018), 2588-2607.  doi: 10.1137/17M1144593.  Google Scholar

[44]

W. LiuM. Röckner and X.-C. Zhu, Large deviation principles for the stochastic quasi-geostrophic equations, Stochastic Process. Appl., 123 (2013), 3299-3327.  doi: 10.1016/j.spa.2013.03.020.  Google Scholar

[45]

W. Liu, C. Tao and J. Zhu, Large deviation principle for a class of SPDE with locally monotone coefficients, Sci. China Math., (2020), In press. doi: 10.1007/s11425-018-9440-3.  Google Scholar

[46]

T. Ma and R.-C. Zhu, Wong-Zakai approximation and support theorem for SPDEs with locally monotone coefficients, J. Math. Anal. Appl., 469 (2019), 623-660.  doi: 10.1016/j.jmaa.2018.09.031.  Google Scholar

[47]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathematical Computation, 13. Chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.  Google Scholar

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